﻿Calculation of Centroids. 249 



is o-(9^?'sin|a rsinja, or ^o-tfr 2 sin 8 %ct. The area o£ the 

 segment is |r 2 « — r 2 sin -|acos J«, and its mass is 



cr/' 2 (-|a — sin \ol cos Ja). 



Since <7</ is v|?'sin-^«, we have 



o-? ,2 (-|« — sin Ja cos \c*)x6 = §o-0?< 3 sin 3 £a, 



or n 



a? = 



?' sin 3 i« 



3 ^a — sin ^acos-^a* 



where a? is the distance of the centroid of the segment from 0. 



Consider next the solids obtained by dividing a right 



circular cylinder into two parts by means of the plane abed 



(fig. 5). Let it be required to find the position of the 



Fis:. 5. 



centroid of the lower solid. We suppose the solid rotated 

 through a small angle 6 about the axis 00' (the axis of 

 figure of the complete cylinder) ; a is thus brought to a', 

 b to b', c to c', and d to d' . In effect the wedge ebb'e'cc' has 

 been removed from the solid and replaced in the position 

 eaa'edd'. If A A denotes an element of area in abed at a 

 distance x from ee\ the volume swept out by this element 

 due to the turning of the solid is AAxO. The mass of this 

 element of volume is p/\Ax0, and since the element is moved 

 (virtually) through a distance "2x, we have, if V is the volume 

 of the solid, 



YpOUe = 2pdZ AAx\ 



where the summation is made over the complete area abed. 

 Hence 



V x OG = AK 2 , 



where A is the area abed, and K is its radius of gyration 



